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// the License.  You may obtain a copy of the License at
//
//      http://www.apache.org/licenses/LICENSE-2.0
//
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// distributed under the License is distributed on an "AS IS" BASIS,
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// limitations under the License.
using Math3.util;
using System;

namespace Math3.special
{
    /// <summary>
    /// This is a utility class that provides computation methods related to the
    /// error functions.
    /// </summary>
    public class Erf
    {
        /// <summary>
        /// The number <c>X_CRIT</c> is used by <see cref="erf(double, double)"/> internally.
        /// This number solves <c>erf(x)=0.5</c> within 1ulp.
        /// More precisely, the current implementations of
        /// <see cref="erf(double)"/> and <see cref="erfc(double)"/> satisfy:<para/>
        /// <c>erf(X_CRIT) < 0.5</c>,<para/>
        /// <c>erf(Math.nextUp(X_CRIT) > 0.5</c>,<para/>
        /// <c>erfc(X_CRIT) = 0.5</c>, and<para/>
        /// <c>erfc(Math.nextUp(X_CRIT) < 0.5</c>
        /// </summary>
        private const double X_CRIT = 0.4769362762044697;

        /// <summary>
        /// Default constructor.  Prohibit instantiation.
        /// </summary>
        private Erf() { }

        /// <summary>
        /// Returns the error function.
        /// <para>erf(x) = 2/&radic;&pi;_0&int;^x e*{-t}_2 dt </para>
        /// <para>This implementation computes erf(x) using the
        /// <see cref="Gamma.regularizedGammaP(double, double, double, int)">regularized gamma 
        /// function</see>,
        /// following <a href="http://mathworld.wolfram.com/Erf.html"> Erf</a>, equation (3)</para>
        /// <para>The value returned is always between -1 and 1 (inclusive).
        /// If <c>abs(x) > 40</c>, then <c>erf(x)</c> is indistinguishable from
        /// either 1 or -1 as a double, so the appropriate extreme value is returned.
        /// </para>
        /// </summary>
        /// <param name="x">the value.</param>
        /// <returns>the error function erf(x)</returns>
        /// <exception cref="MaxCountExceededException">
        /// if the algorithm fails to converge.</exception>
        /// <remarks>See <see cref="Gamma.regularizedGammaP(double, double, double, int)"/></remarks>
        public static double erf(double x)
        {
            if (FastMath.abs(x) > 40)
            {
                return x > 0 ? 1 : -1;
            }
            double ret = Gamma.regularizedGammaP(0.5, x * x, 1.0e-15, 10000);
            return x < 0 ? -ret : ret;
        }

        /// <summary>
        /// Returns the complementary error function.
        /// <para>erfc(x) = 2/&radic;&pi;_x&int;^&infin; e*-t 2 dt
        /// <para/>
        /// = 1 - <see cref="erf(double)"/> </para>
        /// <para>This implementation computes erfc(x) using the
        /// <see cref="Gamma.regularizedGammaQ(double, double, double, int)"/>,
        /// following <a href="http://mathworld.wolfram.com/Erf.html"> Erf</a>, equation (3).</para>
        /// <para>The value returned is always between 0 and 2 (inclusive).
        /// If <c>abs(x) > 40</c>, then <c>erf(x)</c> is indistinguishable from
        /// either 0 or 2 as a double, so the appropriate extreme value is returned.
        /// </para>
        /// </summary>
        /// <param name="x">the value</param>
        /// <returns>the complementary error function erfc(x)</returns>
        /// <exception cref="MaxCountExceededException">
        /// if the algorithm fails to converge.<exception>
        /// <remarks>See <see cref="Gamma.regularizedGammaQ(double, double, double, int)"/></remarks>
        public static double erfc(double x)
        {
            if (FastMath.abs(x) > 40)
            {
                return x > 0 ? 0 : 2;
            }
            double ret = Gamma.regularizedGammaQ(0.5, x * x, 1.0e-15, 10000);
            return x < 0 ? 2 - ret : ret;
        }

        /// <summary>
        /// Returns the difference between erf(x1) and erf(x2).
        /// The implementation uses either erf(double) or erfc(double)
        /// depending on which provides the most precise result.
        /// </summary>
        /// <param name="x1">the first value</param>
        /// <param name="x2">the second value</param>
        /// <returns>erf(x2) - erf(x1)</returns>
        public static double erf(double x1, double x2)
        {
            if (x1 > x2)
            {
                return -erf(x2, x1);
            }

            return
            x1 < -X_CRIT ?
                x2 < 0.0 ?
                    erfc(-x2) - erfc(-x1) :
                    erf(x2) - erf(x1) :
                x2 > X_CRIT && x1 > 0.0 ?
                    erfc(x1) - erfc(x2) :
                    erf(x2) - erf(x1);
        }

        /// <summary>
        /// Returns the inverse erf.
        /// <para>
        /// This implementation is described in the paper:
        /// <a href="http://people.maths.ox.ac.uk/gilesm/files/gems_erfinv.pdf">Approximating
        /// the erfinv function</a> by Mike Giles, Oxford-Man Institute of Quantitative Finance,
        /// which was published in GPU Computing Gems, volume 2, 2010.
        /// The source code is available <a href="http://gpucomputing.net/?q=node/1828">here</a>.
        /// </para>
        /// </summary>
        /// <param name="x">the value</param>
        /// <returns>t such that x = erf(t)</returns>
        public static double erfInv(double x)
        {

            // beware that the logarithm argument must be
            // commputed as (1.0 - x) * (1.0 + x),
            // it must NOT be simplified as 1.0 - x * x as this
            // would induce rounding errors near the boundaries +/-1
            double w = -FastMath.log((1.0 - x) * (1.0 + x));
            double p;

            if (w < 6.25)
            {
                w -= 3.125;
                p = -3.6444120640178196996e-21;
                p = -1.685059138182016589e-19 + p * w;
                p = 1.2858480715256400167e-18 + p * w;
                p = 1.115787767802518096e-17 + p * w;
                p = -1.333171662854620906e-16 + p * w;
                p = 2.0972767875968561637e-17 + p * w;
                p = 6.6376381343583238325e-15 + p * w;
                p = -4.0545662729752068639e-14 + p * w;
                p = -8.1519341976054721522e-14 + p * w;
                p = 2.6335093153082322977e-12 + p * w;
                p = -1.2975133253453532498e-11 + p * w;
                p = -5.4154120542946279317e-11 + p * w;
                p = 1.051212273321532285e-09 + p * w;
                p = -4.1126339803469836976e-09 + p * w;
                p = -2.9070369957882005086e-08 + p * w;
                p = 4.2347877827932403518e-07 + p * w;
                p = -1.3654692000834678645e-06 + p * w;
                p = -1.3882523362786468719e-05 + p * w;
                p = 0.0001867342080340571352 + p * w;
                p = -0.00074070253416626697512 + p * w;
                p = -0.0060336708714301490533 + p * w;
                p = 0.24015818242558961693 + p * w;
                p = 1.6536545626831027356 + p * w;
            }
            else if (w < 16.0)
            {
                w = FastMath.sqrt(w) - 3.25;
                p = 2.2137376921775787049e-09;
                p = 9.0756561938885390979e-08 + p * w;
                p = -2.7517406297064545428e-07 + p * w;
                p = 1.8239629214389227755e-08 + p * w;
                p = 1.5027403968909827627e-06 + p * w;
                p = -4.013867526981545969e-06 + p * w;
                p = 2.9234449089955446044e-06 + p * w;
                p = 1.2475304481671778723e-05 + p * w;
                p = -4.7318229009055733981e-05 + p * w;
                p = 6.8284851459573175448e-05 + p * w;
                p = 2.4031110387097893999e-05 + p * w;
                p = -0.0003550375203628474796 + p * w;
                p = 0.00095328937973738049703 + p * w;
                p = -0.0016882755560235047313 + p * w;
                p = 0.0024914420961078508066 + p * w;
                p = -0.0037512085075692412107 + p * w;
                p = 0.005370914553590063617 + p * w;
                p = 1.0052589676941592334 + p * w;
                p = 3.0838856104922207635 + p * w;
            }
            else if (!Double.IsInfinity(w))
            {
                w = FastMath.sqrt(w) - 5.0;
                p = -2.7109920616438573243e-11;
                p = -2.5556418169965252055e-10 + p * w;
                p = 1.5076572693500548083e-09 + p * w;
                p = -3.7894654401267369937e-09 + p * w;
                p = 7.6157012080783393804e-09 + p * w;
                p = -1.4960026627149240478e-08 + p * w;
                p = 2.9147953450901080826e-08 + p * w;
                p = -6.7711997758452339498e-08 + p * w;
                p = 2.2900482228026654717e-07 + p * w;
                p = -9.9298272942317002539e-07 + p * w;
                p = 4.5260625972231537039e-06 + p * w;
                p = -1.9681778105531670567e-05 + p * w;
                p = 7.5995277030017761139e-05 + p * w;
                p = -0.00021503011930044477347 + p * w;
                p = -0.00013871931833623122026 + p * w;
                p = 1.0103004648645343977 + p * w;
                p = 4.8499064014085844221 + p * w;
            }
            else
            {
                // this branch does not appears in the original code, it
                // was added because the previous branch does not handle
                // x = +/-1 correctly. In this case, w is positive infinity
                // and as the first coefficient (-2.71e-11) is negative.
                // Once the first multiplication is done, p becomes negative
                // infinity and remains so throughout the polynomial evaluation.
                // So the branch above incorrectly returns negative infinity
                // instead of the correct positive infinity.
                p = Double.PositiveInfinity;
            }

            return p * x;

        }

        /// <summary>
        /// Returns the inverse erfc.
        /// </summary>
        /// <param name="x">the value</param>
        /// <returns>t such that x = erfc(t)</returns>
        public static double erfcInv(double x)
        {
            return erfInv(1 - x);
        }
    }
}
